Question

Let O be the origin and R be any point on $y^{2}=2x$. The locus of the midpoint of the line segment OR, is

• A. $(y-1)^{2}=2x$
• B. $y^{2}=3x-1$
• C. $y^{2}+1=2x$
• D. $y^{2}=x$
• E. $x^{2}=2y$

Hint:

Math Tip: To find the locus of a point related to a curve, always assign the moving point coordinates $(h,k)$, relate them to a point $(x_1, y_1)$ on the given curve, and substitute back into the curve's original equation to eliminate $(x_1, y_1)$.

Answer 1

The Correct Option is D

Concept:
Coordinate Geometry - Locus of a Point.
Step 1: Define the coordinates of the points.
The origin is given as $O(0, 0)$.
Let the point $R$ on the given parabola $y^2 = 2x$ have coordinates $(x_1, y_1)$.
Since $R$ lies on the curve, it satisfies the equation: $$ y_1^2 = 2x_1 $$
Step 2: Set up the midpoint formula.
Let $M(h, k)$ be the midpoint of the line segment $OR$.
Using the midpoint formula $M = \left(\frac{x_a+x_b}{2}, \frac{y_a+y_b}{2}\right)$: $$ h = \frac{0 + x_1}{2} \implies x_1 = 2h $$ $$ k = \frac{0 + y_1}{2} \implies y_1 = 2k $$
Step 3: Substitute into the curve's equation.
We need to find the relationship between $h$ and $k$. Substitute the expressions for $x_1$ and $y_1$ from Step 2 into the equation of the parabola from Step 1: $$ (2k)^2 = 2(2h) $$
Step 4: Simplify to find the locus equation.
Expand the squared term and simplify: $$ 4k^2 = 4h $$ Divide both sides by 4: $$ k^2 = h $$
Step 5: Generalize the coordinates.
To express the locus in standard form, replace the specific point $(h, k)$ with general coordinates $(x, y)$: $$ y^2 = x $$